Resolution: Motivation Steps in inferencing (e.g., - PDF document

Resolution: Motivation • Steps in inferencing (e.g., forward-chaining) 1. Define a set of inference rules 2. Define a set of axioms 3. Repeatedly choose one inference rule & one or more axioms (or premices) to derive new sentences until the conclusion sentence is formed • Basic requirement: Rules + axioms should constitute a complete proof system • Observation: Automated inferencing could be a lot more efficient & easy to implement if there was just a single inference rule in the proof system!

Resolution • Resolution (Robinson, 1965): A form of inference that relies on a single rule to prove the truth or falsity of logic sentences • Because of its simplicity, efficiency & completeness properties, resolution has dominated reasoning in AI Key characteristics: • Resolution produces proofs by refutation: “To prove a statement, assume that the negation of the statement is true & try to arrive at a contradiction” • Simplicity achieved by forcing inference rule to operate on sentences that have a very special form called Clause Normal Form (CNF) • Completeness achieved because every logic sentence can be converted to CNF

The Resolution Rule Resolution relies on the following rule: ¬α ⇒ β, β ⇒ γ Resolution rule ¬α ⇒ γ equivalently, α ∨ β, ¬β ∨ γ Resolution rule α ∨ γ Applying the resolution rule: 1. Find two sentences that contain the same literal, once in its positive form & once in its negative form: summer ∨ winter, ¬ winter ∨ cold CNF sentences 2. Use the resolution rule to eliminate the literal from both sentences summer ∨ cold

The Resolution Rule (cont.) A resolution example: at-home ¬ at-home parent clauses: resolvent: empty clause (falsity, contradiction) Another example: ¬ at-home at-home ∨ at-work at-work Observations: • Resolution reduces the length of parent clauses by one literal • Resolution applied after first converting all sentences to CNF form: • Disjunctions only • Negations of atoms only

Resolution in Propositional Logic Basic steps for proving a proposition S: 1. Convert all propositions in premises to CNF p p ¬ (p ∧ q) ∨ r (p ∧ q) ⇒ r ¬ p ∨ ¬ q ∨ r ¬ (s ∨ t) ∨ q ¬ s ∨ q (s ∨ t) ⇒ q ¬ t ∨ q CNF t t t 2. Negate S & convert result to CNF 3. Add negated S to premises 4. Repeat until contradiction or no progress is made: a. Select 2 clauses (call them parent clauses) b. Resolve them together c. If resolvent is the empty clause, a contradiction has been found (i.e., S follows from the premises) d. If not, add resolvent to the premises

Resolution in Propositional Logic Premises: p p (p ∧ q) ⇒ r ¬ p ∨ ¬ q ∨ r (s ∨ t) ⇒ q ¬ s ∨ q ¬ t ∨ q CNF t t A resolution proof of r: ¬ p ∨ ¬ q ∨ r ¬ r p ¬ p ∨ ¬ q ¬ q ¬ t ∨ q ¬ t t

Resolution in First-Order Logic In propopositional logic: at-home ∨ at-work ¬ at-home at-work In first-order logic: ∀ x.at-home(x) ∨ at-work(x) ¬ at-home(y) ? To generalize resolution proofs to FOL we must account for • Predicates • Unbound variables • Existential & universal quantifiers

Clause Form in First-Order Logic • Disjunctions only • Negations of atoms only ¬ P(A,B) • No quantifiers: – universal quantification implicit ∀ x.P(x) → P(x) – existential quantification replaced by Skolem constants/functions ∃ x.P(x) → P(E) ∀ y ∃ x.P(x,y) → P(E(y),y) Ordinary FOL Clause Form none P(A) P(A) ¬¬ elimination ¬¬ Q(A,B) Q(A,B) deMorgan ¬ (P(A) ∧ Q(B,C)) ¬ P(A) ∨ ¬ Q(B,C) deMorgan ¬ (P(A) ∨ Q(B,C)) ¬ P(A) , ¬ Q(B,C) ∧ dropping 2 unit clauses

Clause Form in First-Order Logic Ordinary FOL Clause Form ⇒ elimination P(A) ⇒ Q(B,C) ¬ P(A) ∨ Q(B,C) ⇒ elimination P(A), ¬ Q(B,C) ¬ (P(A) ⇒ Q(B,C)) deMorgan, ∧ drop ∧ drop P(A) ∧ (Q(B,C) ∨ R(D)) P(A), Q(B,C) ∨ R(D) P(A) ∨ Q(B,C), ∧ drop P(A) ∨ (Q(B,C) ∧ R(D)) P(A) ∨ R(D) ∨ distribution ∀ drop ∀ x.P(x) P(x) ⇒ elimination ∀ x.P(x) ⇒ Q(x,A) ¬ P(x) ∨ Q(x,A) ∀ drop P(E) , where E skolemization ∃ x.P(x) is a new constant ⇒ elimination P(A) ⇒ ∃ x.Q(x) ¬ P(A) ∨ Q(F) skolemization deMorgan ¬∀ x.P(x) ¬ P(G) skolemization ∃ x. ¬ P(x)

Clause Form in First-Order Logic Ordinary FOL Clause Form ¬∃ x.P(x) deMorgan ∀ x. ¬ P(x) ¬ P(G) ∀ drop ¬ ( ∃ x.P(x) ∧ ∀ x.Q(x)) variable rename ¬ ( ∃ x.P(x) ∧ ∀ y.Q(y)) deMorgan ¬∃ x.P(x) ∨ ¬∀ y.Q(y) deMorgan ∀ x. ¬ P(x) ∨ ∃ y. ¬ Q(y) ∀ drop ¬ P(x) ∨ ¬ Q(H) skolemization ∀ x ∃ y.P(x,y) fun. skolemization ∀ x.P(x,K(x)) ∀ drop P(x,K(x)) skolemization P(x,y,L(x,y)) ∀ x ∀ y ∃ z.P(x,y,z) ∀ drop

Clause Form in First-Order Logic Ordinary FOL Clause Form ⇒ elimination ¬ P(x) ∨ Q(x,M(x)) ∀ x.P(x) ⇒ ∃ y.Q(x,y) skol., ∀ drop ( ∀ x.P(x)) ⇒ ∃ y.P(y) ⇒ elimination ( ¬∀ x.P(x)) ∨ ∃ y.P(y) deMorgan ∃ x. ¬ P(x) ∨ ∃ y.P(y) ¬ P(N) ∨ P(O) skolemization

Conversion to Clause Form Steps in general case: 1. Rename all variables so that all quantifiers bind distinct variables 2. ⇒ -elimination 3. deMorgan ( ¬∨ , ¬ ∧ , ¬∀ , ¬∃ ) 4. Skolemization ( ∃ -elimination) 5. ∀ -dropping 6. ∨ -distribution 7. ∧ -dropping

Resolution in First-Order Logic In propopositional logic: at-home ∨ at-work ¬ at-home at-work In first-order logic: To generalize resolution proofs to FOL we must account for • Predicates • Unbound variables • Existential & universal quantifiers Idea: First convert sentences to clause form Then unify variables UNIFY at-home(x) ∨ at-work(x) ∀ x.at-home(x) ∨ at-work(x) ¬ at-home(y) ?

Resolution Steps Resolution steps for 2 clauses containing P(arg.list1), ¬ P(arg.list2) 1. Make the variables in the 2 clauses distinct 2. Find the “most general unifier” of arg.list1 & arg.list2: go through the lists “in parallel,” making substitutions for variables only, so as to make the 2 lists the same 3. Make the substitutions corresponding to the m.g.u. throughout both clauses 4. The resolvent is the clause consisting of all the resulting literals except P & ¬ P